Question

From a point 80 feet in front of a library, the angles of elevation to the base of a flagpole, and to the top of the flag are 22° and 43.5°, respectively. Find the height of the flagpole to the nearest foot.

Answer 1

check the picture below.

$\bf tan(22^o)=\cfrac{x}{80}\implies \boxed{80tan(22^o)=x} \\\\[-0.35em] ~\dotfill\\\\ tan(43.5^o)=\cfrac{y+x}{80}\implies 80tan(43.5^o)=y+x\implies 80tan(43.5^o)-x=y \\\\\\ 80tan(43.5^o)-80tan(22^o)=y\implies 43.59506727037783689501 \approx y \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{rounded~up}{44=y}~\hfill$

make sure your calculator is in Degree mode.

Answer 2

Let distance from point of elevation be x

x/80 = tan 22 , x = 80 * tan 22 = 32.32209807 feet

Let distance from bottom of the library to the top of the flag be y

y/80 = tan 43.5 , y = 80 * tan 43.5 = 75.91716534 feet

Height of the flag pole  is y - x = 75.91716534 feet - 32.32209807 feet = 44 feet (to the nearest foot).